SELF-IMPROVING PROPERTIES OF GENERALIZED POINCARÉ TYPE INEQUALITIES THROUGH REARRANGEMENTS

We prove, within the context of spaces of homogeneous type, Lp and exponential type selfimproving properties for measurable functions satisfying the following Poincaré type inequality: inf α ( (f − α)χB )∗ µ ) λµ(B) ≤ cλa(B). Here, f ∗ µ denotes the non-increasing rearrangement of f , and a is a functional acting on balls B, satisfying appropriate geometric conditions. Our main result improves the work in [11], [12] as well as [2], [3] and [14]. Our method avoids completely the “good-λ” inequality technique and any kind of representation formula.